There probably is a God. Many things are easier to explain if there is than if there isn't.

I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematicalideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.

"The Mathematician", in The Works of the Mind (1947) edited by R. B. Heywood, University of Chicago Press, Chicago

Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.

On mistaking pseudorandom number generators for being truly "random" — this quote is often erroneously interpreted to mean that von Neumann was against the use of pseudorandom numbers, when in reality he was cautioning about misunderstanding their true nature while advocating their use. From "Various techniques used in connection with random digits" by John von Neumann in Monte Carlo Method (1951) edited by A.S. Householder, G.E. Forsythe, and H.H. Germond

A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.

"The Role of Mathematics in the Sciences and in Society" (1954) an address to Princeton alumni, published in John von Neumann : Collected Works (1963) edited by A. H. Taub ; also quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by R. Schmalz

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.

It is exceptional that one should be able to acquire the understanding of a process without having previously acquired a deep familiarity with running it, with using it, before one has assimilated it in an instinctive and empirical way… Thus any discussion of the nature of intellectual effort in any field is difficult, unless it presupposes an easy, routine familiarity with that field. In mathematics this limitation becomes very severe.

As quoted in "The Mathematician" in The World of Mathematics (1956), by James Roy Newman

When we talk mathematics, we may be discussing a secondarylanguage built on the primary language of the nervous system.

As quoted in John von Neumann, 1903-1957 (1958) by John C. Oxtoby and B. J. Pettis, p. 128

It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature.

As quoted "John von Neumann (1903 - 1957)" by Eugene Wigner, in Year book of the American Philosophical Society (1958); later in Symmetries and Reflections : Scientific Essays of Eugene P. Wigner (1967), p. 261

You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.

Suggesting to Claude Shannon a name for his new uncertainty function, as quoted in Scientific American Vol. 225 No. 3, (1971), p. 180.

Young man, in mathematics you don't understand things. You just get used to them.

Reply, according to Dr. Felix T. Smith of Stanford Research Institute, to a physicist friend who had said "I'm afraid I don't understand the method of characteristics," as quoted in The Dancing Wu Li Masters: An Overview of the New Physics (1979) by Gary Zukav, Bantam Books, p. 208, footnote.

Statement at the start of a classroom lecture, as quoted in 1,911 Best Things Anyone Ever Said (1988) by Robert Byrne.

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.

As quoted in Bigeometric Calculus: A System with a Scale-Free Derivative (1983) by Michael Grossman, and in Single Variable Calculus (1994) by James Stewart.

With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other.

As quoted in Proportions, Prices, and Planning (1970) by András Bródy

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

Remark made by von Neumann as keynote speaker at the first national meeting of the Association for Computing Machinery in 1947, as mentioned by Franz L. Alt at the end of "Archaeology of computers: Reminiscences, 1945--1947", Communications of the ACM, volume 15, issue 7, July 1972, special issue: Twenty-fifth anniversary of the Association for Computing Machinery, p. 694.

There probably is a God. Many things are easier to explain if there is than if there isn't.

As quoted in John Von Neumann : The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence and Much More (1992) by Norman Macrae, p. 379

If you say why not bomb them tomorrow, I say why not today? If you say today at five o' clock, I say why not one o' clock?

As quoted in "The Passing of a Great Mind" by Clay Blair, Jr., in LIFE Magazine (25 February 1957), p. 96

Bennie decided to approach Johnnie on the matter and arranged to travel to Princeton’s Institute for Advanced Study, headed up at the time by Oppenheimer, where Johnnie (and lesser geniuses such as Albert Einstein) was stationed.

He was a really remarkable man. He listened to me talk about this rather obscure subject and in ten minutes he knew more about it than I did. He was extremely quick.

David Blackwell, as quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz, Mathematical Association of America, ISBN: 9780883855096, p. 213

...throughout much of his career, he led a double life: as an intellectual leader in the ivory tower of pure mathematics and as a man of action, in constant demand as an advisor, consultant and decision-maker to what is sometimes called the military-industrial complex of the United States. My own belief is that these two aspects of his double life, his wide-ranging activities as well as his strictly intellectual pursuits, were motivated by two profound convictions. The first was the overriding responsibility that each of us has to make full use of whatever intellectual capabilities we were endowed with. He had the scientist's passion for learning and discovery for its own sake and the genius's ego-driven concern for the significance and durability of his own contributions. The second was the critical importance of an environment of political freedom for the pursuit of the first, and for the welfare of mankind in general.

The only student of mine I was ever intimidated by. He was so quick. There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

Quite aware that the criteria of value in mathematical work are, to some extent, purely aesthetic, he once expressed an apprehension that the values put on abstract scientific achievement in our present civilization might diminish: "The interests of humanity might change, the present curiosities in science may cease, and entirely different things may occupy the human mind in the future." One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.

A deep sense of humor and an unusual ability for telling stories and jokes endeared Johnny even to casual acquaintances. He could be blunt when necessary, but was never pompous. A mind of von Neumann's inexorable logic had to understand and accept much that most of us do not want to accept and do not even wish to understand. This fact colored many of von Neumann's moraljudgments. … Only scientificintellectualdishonesty and misappropriation of scientific results could rouse his indignation and ire — but these did — and did almost equally whether he himself, or someone else, was wronged.

Eugene Wigner, in "John von Neumann (1903 - 1957)" in Year book of the American Philosophical Society (1958); later in Symmetries and Reflections : Scientific Essays of Eugene P. Wigner (1967), p. 261

I remember a talk that Von Neumann gave at Princeton around 1950, describing the glorious future which he then saw for his computers. Most of the people that he hired for his computer project in the early days were meteorologists. Meteorology was the big thing on his horizon. He said, as soon as we have good computers, we shall be able to divide the phenomena of meteorology cleanly into two categories, the stable and the unstable. The unstable phenomena are those which are upset by small disturbances, the stable phenomena are those which are resilient to small disturbances. He said, as soon as we have some large computers working, the problems of meteorology will be solved. All processes that are stable we shall predict. All processes that are unstable we shall control. He imagined that we needed only to identify the points in space and time at which unstable processes originated, and then a few airplanes carrying smoke generators could fly to those points and introduce the appropriate small disturbances to make the unstable processes flip into the desired directions. A central committee of computer experts and meteorologists would tell the airplanes where to go in order to make sure that no rain would fall on the Fourth of July picnic. This was John von Neumann's dream. This, and the hydrogen bomb, were the main practical benefits which he saw arising from the development of computers.

paraphrase of 1950 von Neumann talk by Freeman Dyson in Infinite in All Directions, 1988; sometimes quoted as "All stable processes we shall predict. All unstable processes we shall control." and attributed to von Neumann directly